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a(n) = floor(n*(2+sqrt(5))^n), equivalently, floor(n*phi^(3n)), where phi = (1+sqrt(5))/2 is the golden ratio.
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%I #48 Jun 14 2022 01:41:18

%S 0,4,35,228,1287,6820,34667,171332,829455,3952836,18604979,86693156,

%T 400623383,1838490212,8387044091,38065809540,171999313951,

%U 774138335108,3472202765123,15525625108324,69229056160039,307921937307684,1366491508589195,6051666872017348

%N a(n) = floor(n*(2+sqrt(5))^n), equivalently, floor(n*phi^(3n)), where phi = (1+sqrt(5))/2 is the golden ratio.

%H Stefano Spezia, <a href="/A354855/b354855.txt">Table of n, a(n) for n = 0..1500</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,-13,-16,13,8,1).

%F a(n) = floor((2+sqrt(5))^n*n).

%F a(n) = floor(n*phi^(3n)) where phi=(1+sqrt(5))/2 is the golden ratio.

%F a(n) = floor(n*F(3n-1)+n*phi*F(3n)), where F(n) = A000045(n) is the n-th Fibonacci number.

%F a(n) = n*L(3n) when n is odd and a(n) = n*L(3n)-1 when n is even (n>=2), where L(n) = A000032(n) is the n-th Lucas number.

%F G.f.: x*(4 + 3*x - 18*x^3 - 4*x^4 - x^5)/((1 - x)*(1 + x)*(1 - 4*x - x^2)^2). - _Stefano Spezia_, Jun 12 2022

%t a[n_] := Floor[n * GoldenRatio^(3*n)]; Array[a, 25, 0] (* _Amiram Eldar_, Jun 09 2022 *)

%Y Cf. A000032, A000045, A001622, A004976, A098317, A128439.

%K nonn,easy

%O 0,2

%A _Jiale Wang_, Jun 09 2022